@ Appl. Gen. Topol. 15, no. 2(2014), 183-202doi:10.4995/agt.2014.2050
c© AGT, UPV, 2014

Function lattices and compactifications

Tomi Alaste

University of Oulu, Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University

of Oulu, Finland (tomi.alaste@gmail.com)

Abstract

Let F be a lattice of real-valued functions on a non-empty set X such

that F contains the constant functions. Using certain filters on X

determined by F, we construct a compact Hausdorff topological space

δX with the property that every bounded member of F extends to

δX and these extensions form a dense subspace of C(δX). If A is any
C

∗
-subalgebra of ℓ

∞(X) containing the constant functions, then our
construction gives a representation of the spectrum of A as a space of

filters on X.

2010 MSC: 46E05; 54D80; 54D35.

Keywords: Function lattice; F-filter; F-ultrafilter; spectrum.

1. Introduction

A widely used method to study topological compactifications and semigroup
compactifications is to view these compactifications as the spectrums of some
C∗-algebras of bounded, complex-valued functions. If X is a Tychonoff space,
then every topological compactification of X can be realized as the spectrum
of some C∗-algebra consisting of continuous functions on X and containing the
constant functions. A similar statement holds for any semigroup compactifica-
tion of a Hausdorff semitopological semigroup S (see [5]).

Some of these compactifications can be considered as spaces of filters. The
most familiar example is the Stone-Čech compactification βX of a discrete
topological space X, which may be regarded as the space of all ultrafilters
on X (see [6] or [9]). If S is a discrete semigroup, then βS is actually a
semigroup compactification of S, and the consideration of βS as the space of
all ultrafilters on S is an extremely powerful approach while analyzing algebraic

Received 17 December 2013 – Accepted 7 May 2014

http://dx.doi.org/10.4995/agt.2014.2050


T. Alaste

properties of βS (see [9]). For a general Tychonoff space X, the Stone-Čech
compactification βX of X can also be considered as a space of filters on X,
but this time one uses z-ultrafilters on X instead of ultrafilters (see [8] or [15]).
(Ultrafilters and z-ultrafilters on X coincide if X is discrete.) The uniform
compactification (or the Samuel compactification, see [10]) of a uniform space
(X,U) was represented as the space of all near ultrafilters on X by Koçak
and Strauss in [12]. Near ultrafilters on X need not be filters in the ordinary
sense of the word, since they need not be closed under finite intersections.
Recently, a representation of the uniform compactification using filters was
given by the author in [1]. In both [12] and [1], the given representation was
used to study the LUC-compactification of a topological group. The LUC-
compactification of a locally compact topological group G was also studied
using filters by Budak and Pym in [3], where the LUC-compactification of G
was considered as a suitable quotient space of the Stone-Čech compactification
of βGd. Here, Gd denotes the group G endowed with the discrete topology.
The WAP-compactification of a discrete semigroup was studied using filters by
Berglund and Hindman in [4] and a treatment of semigroup compactifications
using equivalence classes of z-filters was given by Tootkaboni and Riazi in [14].

The original aim of this paper is to show that the spectrum of any C∗-algebra
F of bounded, complex-valued functions on X, where X is any non-empty set
and F contains the constant functions, can be considered as a space of filters
on X. Since every topological compactification [semigroup compactification]
is determined by the spectrum of some C∗-algebra of bounded functions, our
development gives a unified treatment of all these compactifications as spaces
of filters. As far as we are aware, for many C∗-algebras our approach is actu-
ally the first one using filters instead of equivalence classes of filters or quotient
spaces of some other compactifications. If X is a discrete topological space,
then our approach yields the usual representation of βX as the space of all ul-
trafilters on X. Independently of the C∗-algebra F in question, our approach
has a number of similarities with the consideration of βX for a discrete topo-
logical space X as the space of all ultrafilters on X. For example, we obtain
a bijective correspondence between non-empty, closed subsets of the spectrum
of F and F-filters on X. We believe that the method presented in this paper
can serve as a valuable tool in the study of both topological compactifications
and semigroup compactifications. This method was used by the author in [1]
to study the smallest ideal of the LUC-compactification of a topological group
and in [2] to study the smallest ideal of any semigroup compactification of any
semitopological semigroup.

For a large part of the theory developed in this paper, it is not necessary
that we work with a C∗-algebra of bounded functions. Instead, it is the lat-
tice structure of real-valued functions that is important for our development.
Therefore, we work with a lattice of real-valued functions (which might contain
unbounded functions) throughout Sections 3-8. In Section 3, we introduce the
main object of this paper, namely F-filters and F-ultrafilters, and we study
some of their basic properties. In Section 4, we define a topology on the set

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Function lattices and compactifications

of all F-ultrafilters and we show that the resulting space δX is a compact
Hausdorff space. Furthermore, we show that the F-filters describe the topol-
ogy of δX in a similar way as filters describe the topology of the Stone-Čech
compactification of a discrete topological space. Section 5 contains a study
of continuous functions on δX. We show that every bounded member of F
extends to δX and that these extensions form a dense subspace of the algebra
of all continuous, real-valued functions on δX. It is remarkable that we do not
need the Stone-Weierstrass Theorem to prove the density of these extensions.
In Sections 6-8, our main results concern closed subalgebras of the algebra of
all bounded, real-valued functions on X. In Section 7, we establish a corre-
spondence between F-filters and closed, proper ideals of F. Section 8 contains
a treatment of F-filters on a Hausdorff topological space X in the case that
every member of F is a continuous function on X. In the last section, we turn
our attention to C∗-algebras of bounded, complex-valued functions. Here, we
include a description how the developed theory so far can be used to produce
an interpretation of the spectrum of such an algebra [compactification of X] as
a space of filters on X.

Our construction of the space δX as the space of all F-ultrafilters has some
similarities with the consideration of the Smirnov compactification of a prox-
imity space using maximal round filters (see [13]). If the function lattice F
on a non-empty set X separates the points of X, then there is a bijective
correspondence between F-ultrafilters on X and maximal round filters on the
proximity space (X,P), where P is the proximity on X generated by F. An
advantage of our construction is that it applies to any function lattice F on X,
and so it applies also to those semigroup compactifications where the evalua-
tion mapping is not necessarily injective. This includes, for example, the Bohr
compactification of some topological groups.

2. Preliminaries

Throughout the paper, let X be any non-empty set. We denote by F(X) the
algebra of all real-valued functions on X. We denote by ℓ∞(X) the subalgebra
of F(X) consisting of all bounded members of F(X). Recall that the space
ℓ∞(X) is equipped with the norm of uniform convergence. A function f ∈ F(X)
is positive if and only if f(x) ≥ 0 for every x ∈ X. For all f,g ∈ F(X), the
functions (f ∨ g) : X → R and (f ∧ g) : X → R are defined by

(f ∨ g)(x) = max{f(x),g(x)} and (f ∧ g)(x) = min{f(x),g(x)}

for every x ∈ X, respectively. By a function lattice on X we mean a vector
subspace F of F(X) such that F contains the constant functions and f ∨g ∈ F
and f ∧ g ∈ F for all f,g ∈ F. Note that a vector subspace F of F(X) is a
function lattice on X if and only if |f| ∈ F for every f ∈ F.

We denote by N the set of all positive integers, that is, N = {1,2,3, . . .}. We
denote by P(X) the family of all subsets of X. A filter on X is a non-empty
family ϕ of subsets of X with the following properties:

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T. Alaste

(i) If A,B ∈ ϕ, then A ∩ B ∈ ϕ.
(ii) If A ∈ ϕ and A ⊆ B ⊆ X, then B ∈ ϕ.
(iii) ∅ /∈ ϕ.

A filter base on X is a non-empty family B of subsets of X such that ∅ /∈ B
and, for all sets A,B ∈ B, there exists some C ∈ B such that C ⊆ A ∩ B. If B
is a filter base on X, then the filter ϕ on X generated by B is

ϕ = {A ⊆ X : there exists some B ∈ B such that B ⊆ A}.

Let ϕ be a filter on X. A family B of subsets of X is a filter base for ϕ if and
only if B ⊆ ϕ and, for every A ∈ ϕ, there exists some B ∈ B such that B ⊆ A.

Let (Y,τ) be a (not necessarily Hausdorff) topological space. For every
subset A of Y , we denote by int(Y,τ)(A) and cl(Y,τ)(A) the interior and the
closure of A in (Y,τ), respectively, or simply by intY (A) and clY (A) if τ is
understood. We denote by C(Y ) the subalgebra of ℓ∞(X) consisting of all
continuous members of ℓ∞(X). If Y is locally compact, then the subalgebra
C0(X) of C(X) consists of those members of C(X) which vanish at infinity.

3. F-filters

Throughout this section, let F be a function lattice on X. We introduce
the main object of the paper, namely F-filters and F-ultrafilters on X, and we
describe some of their basic properties. For all f ∈ F and r > 0, we put

Z(f) = {x ∈ X : f(x) = 0} and X(f,r) = {x ∈ X : |f(x)| ≤ r}.

Definition 3.1. An F-family on X is a non-empty family A of non-empty
subsets of X such that, for every A ∈ A with A 6= X, there exist some B ∈ A
and a function f ∈ F such that f(B) = {0} and f(X \ A) = {1}. An F-filter
on X is a filter ϕ on X which is also an F-family on X.

Since F contains the constant functions, we may assume that the function
f ∈ F in the previous definition satisfies f(B) = {1} and f(X \ A) = {0}.
Also, since F is closed under the lattice operations ∨ and ∧, we may assume,
if necessary, that f(X) ⊆ [0,1].

There exists at least one F-filter on X, namely the filter ϕ = {X}. If F
contains only the constant functions, then {X} is the only F-filter on X. On
the other hand, if F = ℓ∞(X), then every filter ϕ on X is an F-filter on X.

Let ϕ be an F-filter on X and suppose that A ∈ ϕ satisfies A 6= X. Pick
some B ∈ ϕ and a function f ∈ F with f(B) = {0} and f(X \A) = {1}. Then
B ⊆ Z(f) ⊆ A. Since Z(f) ∈ ϕ, the filter ϕ has a filter base consisting of zero
sets (determined by F) of X. However, not every zero set of X is contained in
any F-filter. For example, let F = C(R). Then A = {0} is a zero set of R but
there is no F-filter ϕ on R satisfying A ∈ ϕ.

We shall apply the following remark frequently without any further notice.

Remark 3.2. Let A be a non-empty family of non-empty subsets of X. Suppose
that, for every A ∈ A with A 6= X, there exist some B ∈ A, real numbers s
and r with s < r, and a function f ∈ F such that f(x) ≤ s for every x ∈ B

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Function lattices and compactifications

and f(x) ≥ r for every x ∈ X \A. Then, using the lattice operations, it is easy
to see that A is an F-family on X.

Zorn’s Lemma implies that every F-filter on X is contained in some maximal
(with respect to inclusion) F-filter on X.

Definition 3.3. An F-ultrafilter on X is an F-filter on X which is not properly
contained in any other F-filter on X.

Note that if F = ℓ∞(X), then a filter ϕ on X is an F-ultrafilter if and only
if ϕ is an ultrafilter on X. Also, the following fact about F-ultrafilters is very
useful: If p and q are F-ultrafilters on X, then p = q if and only if p ⊆ q.

Definition 3.4. Define

F0 = {f ∈ F : X(f,r) 6= ∅ for every r > 0}.

For every non-empty subset A of X, define

Z(A) = {f ∈ F : f(x) = 0 for every x ∈ A}.

The next statement follows from Remark 3.2.

Lemma 3.5. The family A = {X(f,r) : f ∈ F′, r > 0} is an F-family on X
for every non-empty subset F′ of F0.

We will use the following lemma and its corollaries a number of times in
this paper. Recall that a non-empty family A of subsets of X has the finite
intersection property if and only if

⋂n
k=1 Ak 6= ∅ whenever A1, . . . ,An ∈ A for

some n ∈ N.

Lemma 3.6. If A is an F-family on X such that A has the finite intersection
property, then there exists an F-ultrafilter p on X such that A ⊆ p.

Proof. We sketch the proof briefly. Let ϕ be the smallest filter on X containing
the family A. Let n ∈ N and suppose that A1, . . .An ∈ A satisfy Ak 6= X for
every k ∈ {1, . . . ,n}. If k ∈ {1, . . . ,n}, then there exist some Bk ∈ A and
a positive function fk ∈ F with fk(Bk) = {0} and fk(X \ Ak) = {1}. Put
B =

⋂n
k=1 Bk and f =

∑n
k=1 fk. Since B ∈ ϕ, f ∈ F, f(B) = {0}, and

f(x) ≥ 1 for every x ∈ X \
⋂n

k=1 Ak, the filter ϕ is an F-filter on X. �

The next two corollaries now follow from Lemma 3.5.

Corollary 3.7. Let ϕ be an F-filter on X and let f ∈ F. If X(f,r) ∩ B 6= ∅
for every B ∈ ϕ and for every r > 0, then there exists an F-ultrafilter p on X
such that ϕ ∪ {X(f,r) : r > 0} ⊆ p.

Corollary 3.8. Let ϕ be an F-filter on X and let A ⊆ X. If A ∩ B 6= ∅ for
every B ∈ ϕ, then there exists an F-ultrafilter p on X containing the family
ϕ ∪ {X(f,r) : f ∈ Z(A), r > 0}.

If F = ℓ∞(X), then we may take the members of F in the next theorem
to be characteristic functions of subsets of X. Then, except for statement (ii),
the conclusion of the next theorem is the same as in [9, Theorem 3.6].

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T. Alaste

Theorem 3.9. If ϕ ⊆ P(X), then the following statements are equivalent:

(i) ϕ is an F-ultrafilter on X.
(ii) ϕ is an F-filter on X and, if X(f,r) /∈ ϕ for some f ∈ F and r > 0,

then, for every real number t with 0 < t < r, there exists some A ∈ ϕ
such that X(f,t) ∩ A = ∅.

(iii) ϕ is a maximal F-family on X such that ϕ has the finite intersection
property.

(iv) ϕ is an F-filter on X and, if
⋃n

k=1 Ak ∈ ϕ for some n ∈ N and for some
A1, . . . ,An ⊆ X, then there exists k ∈ {1, . . . ,n} such that X(f,r) ∈ ϕ
for all f ∈ Z(Ak) and r > 0.

(v) ϕ is an F-filter on X and, if A ⊆ X satisfies A 6= ∅ and A 6= X, then,
either X(f,r) ∈ ϕ for every f ∈ Z(A) and for every r > 0, or X(g,r) ∈ ϕ
for every g ∈ Z(X \ A) and for every r > 0.

Proof. (i) ⇒ (ii) This follows from Corollary 3.8 with g = (|f| − t) ∨ 0. Note
that g ∈ Z(X(f,t)) and X(g,r − t) ⊆ X(f,r).

(ii) ⇒ (iii) This follows from the definition of an F-family.
(iii) ⇒ (iv) Suppose that (iii) holds. Let us first show that ϕ is a filter on X.

Clearly, X ∈ ϕ, ∅ /∈ ϕ, and B ∈ ϕ whenever A ∈ ϕ and A ⊆ B ⊆ X. So, let
A,B ∈ ϕ. Pick some C,D ∈ ϕ and functions f,g ∈ F with f(C) = g(D) = {0}
and f(X \ A) = g(X \ B) = {1}. Since C ∩ D ⊆ X(|f| + |g|,r) for every
r > 0, we have X(|f| + |g|,r) ∈ ϕ for every r > 0 by Lemma 3.5. Since
X(|f| + |g|,1/2) ⊆ A ∩ B, we have A ∩ B ∈ ϕ, as required.

Suppose now that
⋃n

k=1 Ak ∈ ϕ for some n ∈ N and for some non-empty
subsets A1, . . . ,An of X. Suppose also that, for every k ∈ {1, . . . ,n}, there exist
rk > 0 and a function fk ∈ Z(Ak) such that X(fk,rk) /∈ ϕ. If k ∈ {1, . . . ,n},
then the family A = ϕ∪ {X(fk, t) : t > 0} is an F-family on X by Lemma 3.5.
Since A contains ϕ properly, there exist some Bk ∈ ϕ and tk > 0 such that
Bk∩X(fk, tk) = ∅. Put B =

⋂n
k=1 Bk. Then B ∈ ϕ and B∩[

⋃n
k=1 Z(fk)] = ∅,

a contradiction.
(iv) ⇒ (v) This is obvious.
(v) ⇒ (i) Suppose that (v) holds. Suppose also that there exists an F-filter

ψ on X which properly contains ϕ. Pick some set A ∈ ψ \ ϕ. Pick some
B ∈ ψ and a function f ∈ F with f(B) = {0} and f(X \ A) = {1}. Pick
some C ∈ ψ and a function g ∈ F with g(C) = {1} and g(X \ B) = {0}. Since
X(f,1/2) ⊆ A, we have X(f,1/2) /∈ ϕ. Since f ∈ Z(B), we have X(g,1/2) ∈ ϕ
by assumption. But now X(g,1/2) ∩ C = ∅, a contradiction. �

The two statements given in statement (v) of the previous theorem are not
exclusive. Indeed, let F = C(R) and A = Q. Then Z(A) = Z(R \ A) = {0},
and so X(f,r) = X(g,r) = X for all f ∈ Z(A), g ∈ Z(R \ A), and r > 0.

4. The topological space δX

As in the previous section, we assume that F is a function lattice on X.
Our next task is to define a topology on the set of all F-ultrafilters on X

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Function lattices and compactifications

and establish some of the properties of the resulting space. In particular, we
show that the resulting space is a compact Hausdorff space and that F-filters
describe its topology.

Definition 4.1. Define δX = {p : p is an F-ultrafilter on X}. For every sub-

set A of X, put  = {p ∈ δX : A ∈ p}. For every F-filter ϕ on X, put
ϕ̂ = {p ∈ δX : ϕ ⊆ p}.

To be precise, we should include the function lattice F in the notation above,
such as δF(X). Except in Section 6, we consider only one function lattice F in
the same context, so we hope that the notation chosen above does not cause
any misunderstandings.

Theorem 4.2. If ϕ and ψ are F-filters on X, then the following statements
hold:

(i) ϕ̂ =
⋂

A∈ϕ
Â.

(ii) ϕ =
⋂

p∈ϕ̂
p.

(iii) ϕ ⊆ ψ if and only if ψ̂ ⊆ ϕ̂.

(iv) ϕ = ψ if and only if ϕ̂ = ψ̂.

Proof. (i) This is obvious.
(ii) The inclusion ϕ ⊆

⋂
p∈ϕ̂

p is obvious, so suppose that A is a subset of

X such that A /∈ ϕ. By Corollary 3.8, there exists an element p ∈ ϕ̂ such that
{X(f,r) : f ∈ Z(X \ A), r > 0} ⊆ p. Now, it is enough to show that A /∈ p.
Suppose that A ∈ p. Pick some B ∈ p and a function f ∈ F with f(B) = {1}
and f(X \ A) = {0}. Since f ∈ Z(X \ A), we have X(f,1/2) ∈ p. But now
B ∩ X(f,1/2) = ∅, a contradiction.

(iii) Necessity is obvious and sufficiency follows from statement (ii).
(iv) This follows from statement (iii). �

The family {Â : A ⊆ X} is a base for a topology on δX. We define the
topology of δX to be the topology which has this family as its base. In partic-

ular, {Â : A ∈ p} is a neighborhood base of a point p ∈ δX. If Y ⊆ δX, then

we denote clδX(Y ) by Y with one exception: If A ⊆ X, then we use clδX(Â)

instead of the cumbersome notation Â.
We denote by τ(F) the weakest topology τ on X such that every member

of F is continuous with respect to τ. For every subset A of X, we denote
int(X,τ(F))(A) by A

◦. For every element x ∈ X, we denote by NF(x) the
neighborhood filter of x in (X,τ(F)).

We shall apply the following remark frequently without any further notice.

Remark 4.3. Let ϕ be an F-filter on X. Suppose that A ∈ ϕ satisfies A 6= X.
Pick some B ∈ ϕ and a function f ∈ F with f(B) = {0} and f(X \ A) = {1}.
Then B ⊆ {x ∈ X : |f(x)| < 1} ⊆ A. In conclusion, if C is any subset of X,
then C ∈ ϕ if and only if C◦ ∈ ϕ.

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T. Alaste

Theorem 4.4. If x ∈ X, then the family

Ax = {X(f,r) : f ∈ F, f(x) = 0, and r > 0}

is a filter base on X. The filter on X generated by Ax is NF(x) and it is an
F-ultrafilter on X.

Proof. If f,g ∈ F and r > 0, then X(|f| + |g|,r) ⊆ X(f,r) ∩ X(g,r). This
implies that Ax is a filter base on X. Clearly, Ax generates the filter NF(x),
and so NF(x) is an F-filter on X by Lemma 3.5. Then NF(x) is an F-ultrafilter
on X by Theorem 3.9 (iv). �

The following definition is reasonable by the previous theorem.

Definition 4.5. The function e : X → δX defined by e(x) = NF(x) for every
x ∈ X is the canonical mapping.

If A ⊆ X and x ∈ X, then e(x) ∈ Â if and only if x ∈ A◦. Next, let

A,B ⊆ X. In general, B̂ ∩ e(A) = ∅ does not imply B ∩ A = ∅. However,
this implication holds if B is a τ(F)-open subset of X. We apply this fact
repeatedly in what follows.

We gather some properties of the space δX in the following lemmas.

Lemma 4.6. Let A ⊆ X and let p ∈ δX. The following statements are
equivalent:

(i) p ∈ e(A).
(ii) A ∩ B 6= ∅ for every B ∈ p.
(iii) X(f,r) ∈ p for every f ∈ Z(A) and for every r > 0.

In particular, p ∈ e(A) for every A ∈ p.

Proof. (i) ⇒ (ii) If A ∩ B = ∅ for some B ∈ p, then A ∩ B◦ = ∅, and so

e(A) ∩ B̂◦ = ∅. Since B◦ ∈ p, we have p /∈ e(A).
(ii) ⇒ (iii) This follows from Corollary 3.8.

(iii) ⇒ (i) Suppose that p /∈ e(A). Then there exists a τ(F)-open subset B

of X such that B ∈ p and B̂ ∩ e(A) = ∅, and so B ∩ A = ∅. Pick some C ∈ p
and a function f ∈ F with f(C) = {1} and f(X \ B) = {0}. Then f ∈ Z(A).
Since X(f,1/2) ∩ C = ∅, we have X(f,1/2) /∈ p, and so statement (iii) does
not hold. �

Lemma 4.7. If A,B ⊆ X, then the following statements hold:

(i) X̂ \ A = δX \ e(A).

(ii) If A is a τ(F)-open subset of X, then e(A) = clδX(Â).

(iii) Â = B̂ if and only if A◦ = B◦.

(iv) Â = ∅ if and only if A◦ = ∅.

(v) Â = δX if and only if A = X.

Proof. (i) Suppose first that p ∈ X̂ \ A. Since X̂ \ A ∩ e(A) = ∅, we have

p /∈ e(A). Suppose now that p ∈ δX \ e(A). Then there exists a τ(F)-open

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Function lattices and compactifications

subset C of X such that C ∈ p and Ĉ ∩ e(A) = ∅. Then C ∩ A = ∅, that is,
C ⊆ X \ A, and so X \ A ∈ p.

(ii) The inclusion clδX(Â) ⊆ e(A) holds for any subset A of X and follows

from statement (i). Suppose now that A is τ(F)-open and that p ∈ e(A). If

B ∈ p, then B̂ ∩ e(A) 6= ∅, so B◦ ∩ A 6= ∅, and so B̂ ∩ Â 6= ∅. Therefore,

p ∈ clδX(Â).
Statement (iii) follows from Remark 4.3. Then statements (iv) and (v) follow

from statement (iii). �

Lemma 4.8. Let A,B ⊆ X. Then X(f,r) ∩ X(g,r) 6= ∅ for all f ∈ Z(A),

g ∈ Z(B), and r > 0 if and only if e(A) ∩ e(B) 6= ∅.

Proof. Necessity follows from Lemma 4.6, so suppose that X(f,r)∩X(g,r) 6= ∅
for all f ∈ Z(A), g ∈ Z(B), and r > 0. Put

A = {X(h,r) : h ∈ Z(A) ∪ Z(B), r > 0}.

Then A is an F-family on X by Lemma 3.5. We claim that A has the finite
intersection property. Let f1, . . . ,fn ∈ Z(A) and let g1, . . . ,gm ∈ Z(B) for
some n,m ∈ N. Then f :=

∑n
k=1|fk| ∈ Z(A) and g :=

∑m
k=1|gk| ∈ Z(B). If

r > 0, then

X(f,r) ∩ X(g,r) ⊆

(
n⋂

k=1

X(fk,r)

)
∩

(
m⋂

k=1

X(gk,r)

)
,

thus verifying our claim. By Lemma 3.6, there exists an element p ∈ δX such
that A ⊆ p. Then p ∈ e(A) ∩ e(B) by Lemma 4.6, as required. �

Now, we are ready to prove first of the main theorems of this section.

Theorem 4.9. The space δX is a compact Hausdorff space and e(X) is dense
in δX.

Proof. First, e(X) is dense in δX by Lemma 4.7 (iv). To see that δX is
Hausdorff, let p and q be distinct points of δX. Pick some set A ∈ p \ q. Pick
some B ∈ p and a function f ∈ F with f(B) = {0} and f(X \ A) = {1}.
Since X(f,1/2) /∈ q, there exists some C ∈ q such that X(f,1/3) ∩ C = ∅ by

Theorem 3.9 (ii). Then B ∩C = ∅, and so B̂ and Ĉ are disjoint neighborhoods
of p and q, respectively.

Lemma 4.7 (i) implies that the family B = {e(A) : A ⊆ X} is a base for the
closed subsets of δX. Suppose that a subset C of B has the finite intersection
property. To show that δX is compact, it is enough to show that

⋂
C∈C

C 6= ∅.

Put A′ = {A ⊆ X : e(A) ∈ C} and A = {X(f,r) : A ∈ A′, f ∈ Z(A), r > 0}.
Then A is an F-family on X by Lemma 3.5 and A has the finite intersection
property by Lemma 4.8. By Lemma 3.6, there exists an element p ∈ δX such
that A ⊆ p. Then p ∈ e(A) for every A ∈ A′ by Lemma 4.6, and so p ∈

⋂
C∈C

C,
thus finishing the proof. �

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 191



T. Alaste

We finish this section by showing that F-filters describe the topology of δX.
As with the Stone-Čech compactification of a discrete topological space, we
have two natural candidates for the closure of an F-filter in δX, namely ϕ̂ and
the following.

Definition 4.10. Define ϕ =
⋂

A∈ϕ
e(A) for every F-filter ϕ on X.

Note that ϕ is a non-empty, closed subset of δX. The next statement follows
from Lemma 4.6.

Theorem 4.11. If ϕ is an F-filter on X, then ϕ̂ = ϕ.

Theorem 4.12. If C is a non-empty, closed subset of δX, then there exists a
unique F-filter ϕ on X such that ϕ̂ = C.

Proof. Let C be a non-empty, closed subset of δX. Put ϕ =
⋂

p∈C
p. Clearly,

ϕ is a filter on X. Let us show that ϕ is an F-family on X, hence, an F-filter
on X. Suppose that A ∈ ϕ satisfies A 6= X. If p ∈ C, then A ∈ p, and so there
exist some Bp ∈ p and a function fp ∈ F with f(X) ⊆ [0,1], fp(Bp) = {0},

and fp(X \ A) = {1}. Now, {B̂p : p ∈ C} is an open cover of C, and so there

exist points p1, . . . ,pn ∈ C for some n ∈ N such that C ⊆
⋃n

k=1 B̂pk . Put
f =

∑n
k=1 fpk and B =

⋃n
k=1 Bpk . Then B ∈ ϕ. Since f(x) ≤ n − 1 for every

x ∈ B and f(x) = n for every x ∈ X \ A, the filter ϕ is an F-family on X.
Let us verify the equality ϕ̂ = C. The inclusion C ⊆ ϕ̂ is obvious, so suppose

that q ∈ δX \C. Then there exists a τ(F)-open subset A of X such that A ∈ q

and  ∩ C = ∅. For every p ∈ C, pick a τ(F)-open subset Bp of X such that

Bp ∈ p and  ∩ B̂p = ∅. Then A ∩ Bp = ∅ for every p ∈ C. As above, there
exist n ∈ N and points p1, . . . ,pn ∈ C such that B :=

⋃n
k=1 Bpk ∈ ϕ. Since

A ∩ B = ∅, we have q /∈ ϕ̂, as required.
Finally, the F-filter ϕ on X satisfying ϕ̂ = C is unique by Theorem 4.2

(iv). �

5. Continuous functions on δX

Again, we assume that F is a function lattice on X. This section is devoted
to a study of continuous, real-valued functions on the space δX. We show that
every bounded member of F extends to δX and that these extensions form a
dense subspace of C(δX).

We leave the proof of the following lemma to the reader.

Lemma 5.1. Let p ∈ δX, let g ∈ C(δX), and let r > 0. Then

{x ∈ X : |g(p) − g(e(x))| ≤ r} ∈ p.

Theorem 5.2. For every bounded function f ∈ F, there exists a unique func-
tion f̂ ∈ C(δX) satisfying f = f̂ ◦ e.

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Function lattices and compactifications

Proof. Let p ∈ δX and put

(5.1) C =
⋂

A∈p

clR(f(A)).

Then C is a non-empty subset of R by assumption. Choosing any element

f̂(p) ∈ C, we obtain a function f̂ : δX → R.
Next, let us show that if p = e(x) for some x ∈ X, then C = {f(x)}. This

will establish the equality f = f̂ ◦ e. Clearly, f(x) ∈ C, so let y ∈ R be such
that y 6= f(x). Pick r > 0 such that y /∈ U := [f(x) − r,f(x) + r]. Then
f−1(U) ∈ e(x). Since y /∈ clR(f(f

−1(U))), we have y /∈ C, as required.
The density of e(X) in δX implies that the continuous function h on δX

satisfying f = h ◦ e is unique. Therefore, it is enough to show that f̂ is
continuous to finish the proof. To see that f̂ is continuous, let p ∈ δX and put
g = f −f̂(p). First, we claim that X(g,r) ∈ p for every r > 0. By Corollary 3.7,
it is enough to show that X(g,r) ∩ B 6= ∅ for every B ∈ p and for every r > 0.

So, let B ∈ p and r > 0 be given. Since f̂(p) ∈ clR(f(B)), there exists a point

x ∈ B such that |g(x)| = |f(x)− f̂(p)| ≤ r, and so x ∈ X(g,r)∩B, as required.

To finish the proof, let r > 0. If q ∈ X̂(g,r), then f̂(q) ∈ clR(f(X(g,r))), so

|f̂(q) − f̂(p)| ≤ r, and so f̂ is continuous at p. �

Although the canonical mapping need not be injective, we call the continuous

function f̂ on δX satisfying f = f̂ ◦ e an extension of f to δX.
We denote by R∗ the one-point compactification of R, that is, R∗ = R∪{∞}.

Theorem 5.3. For every function f ∈ F, there exists a unique continuous
function f̂ : δX → R∗ satisfying f = f̂ ◦ e.

Proof. Arguing as in the previous proof and using the compactness of R∗, we

need only to show that f̂ is continuous at a point p ∈ δX with f̂(p) = ∞. Let
n ∈ N. By Lemma 4.6, it is enough to show that A := {x ∈ X : |f(x)| ≥ n} ∈ p.
Put B = {x ∈ X : |f(x)| ≥ n + 1} and g = n + 1 − (|f| ∧ (n + 1)). Then

g ∈ Z(B) and X(g,1) ⊆ A. Since p ∈ e(B), we have A ∈ p by Lemma 4.6, as
required. �

The points p ∈ δX satisfying f̂(p) = ∞ have a simple characterization.
Indeed, if f ∈ F is unbounded, then the sets Cn = {x ∈ X : |f(x)| ≥ n}, where
n ∈ N, determine a filter base B on X. The filter ϕ on X generated by B is an

F-filter on X and satisfies ϕ̂ = {p ∈ δX : f̂(p) = ∞}.
In the next theorem (and later), we put Fb = F ∩ ℓ

∞(X). Recall that the
space Fb is equipped with the norm of uniform convergence. We could deduce
the following theorem from the Stone-Weierstrass Theorem. However, we feel
that the proof below is worth presenting, since it uses only properties of F-
filters instead of the Stone-Weierstrass Theorem. Also, in this way we obtain
the Stone-Weierstrass Theorem as a corollary in Section 8.

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T. Alaste

Theorem 5.4. The mapping Γ : Fb → C(δX) defined by Γ(f) = f̂ is a linear
isometry and Γ(F) is dense in C(δX). If Fb is an algebra, then Γ is also a
homomorphism.

Proof. Using the density of e(X) in δX and the equality f = f̂ ◦ e, it is easy
to verify that Γ is a linear isometry (homomorphism if Fb is an algebra) and
we leave the details to the reader. Let us show that Γ(F) is dense in C(δX).
To prove this, it is enough to show that, for every positive function g ∈ C(δX)
with ‖g‖ = 1 and for every r > 0, there exists a function f ∈ Fb such that

‖f̂ − g‖≤ r. So, let g ∈ C(δX) be positive with ‖g‖ = 1 and let r > 0. Pick
n ∈ N such that 1/n ≤ r/3. For every k ∈ {1, . . . ,n}, define the following
subsets of [0,1], X, and δX, respectively:

Ik =
[k − 1

n
,
k

n

]
, Ak = {x ∈ X :

k − 2

n
< g(e(x)) <

k + 1

n
}, Ck = g

−1(Ik).

Note that Ak ∩ Aj = ∅ whenever k,j ∈ {1, . . . ,n} and k + 3 ≤ j.
Let k ∈ {1, . . . ,n}. If p ∈ Ck, then Ak ∈ p by Lemma 5.1, and so there

exist some Bp ∈ p and fp ∈ Fb with fp(Bp) = {k/n}, fp(X \ Ak) = {0}, and
fp(X) ⊆ [0,k/n]. Pick elements p1, . . . ,pm ∈ Ck for some m ∈ N such that

Ck ⊆
⋃m

j=1 B̂pj and put fk = fp1 ∨ . . . ∨ fpm. Note that fk(X \ Ak) = {0} and

fk(x) = k/n for every x ∈ X with e(x) ∈ Ck.

Put f = f1 ∨. . .∨fn. Then f ∈ Fb and we claim that ‖f̂ −g‖ ≤ r. To verify
our claim, it is enough to show that |f(x) − g(e(x))| ≤ r for every x ∈ X. So,
let x ∈ X. Suppose first that g(e(x)) ≥ (n − 3)/n. Then e(x) ∈ Ck for some
k ∈ N with n − 2 ≤ k ≤ n, so f(x) ≥ (n − 2)/n, and so |f(x) − g(e(x))| ≤ r.
Suppose now that g(e(x)) < (n − 3)/n. Then there exists k ∈ {1, . . . ,n − 3}
such that (k − 1)/n ≤ g(e(x)) < k/n. Then x ∈ Ak and e(x) ∈ Ck, and so
f(x) ≥ k/n. Since Ak ∩Aj = ∅ for every j ∈ {1, . . . ,n} with j ≥ k+3, we have
fj(x) = 0 for every j with k + 3 ≤ j ≤ n, and so f(x) ≤ (k + 2)/n. Therefore,
|f(x) − g(e(x))| ≤ 3/n ≤ r, thus finishing the proof. �

Any closed subalgebra of ℓ∞(X) containing the constant functions is a func-
tion lattice on X (see [16, p. 291] or [11, p. 265]). Therefore, we obtain the
following corollary.

Corollary 5.5. If F is a closed subalgebra of ℓ∞(X) containing the constant
functions, then Γ : F → C(δX) is an isometric isomorphism.

Corollary 5.6. If F is a function lattice on X such that F ⊆ ℓ∞(X), then
the closure of F in ℓ∞(X) is a closed subalgebra of ℓ∞(X).

Proof. Denote by F′ the closure of F in ℓ∞(X). Remark 3.2 implies that a
filter ϕ on X is an F-filter if and only if ϕ is an F′-filter, and so the notation δX
is unambiguous. Corollary 5.5 implies that the mapping Γ : F′ → C(δX) is an
isometric isomorphism. Since C(δX) is an algebra, the statement follows. �

Next, we show that F-filters describe all dense images of X in compact
Hausdorff spaces. Precise statement and details follow.

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Function lattices and compactifications

Theorem 5.7. Let Y be a compact Hausdorff space and let ε : X → Y be a
function such that ε(X) is dense in Y . The following statements hold:

(i) The set F = {h◦ε : h ∈ C(Y )} is a closed subalgebra of ℓ∞(X) containing
the constant functions.

(ii) F is isometrically isomorphic with C(Y ).
(iii) There exists a homeomorphism F : δX → Y such that F ◦ e = ε.

Proof. We prove only statement (iii) and leave the verifications of statements
(i) and (ii) to the reader. If p ∈ δX, then

C =
⋂

A∈p

clY (ε(A))

is a non-empty subset of Y , and we claim that C is a singleton. Suppose that
C contains distinct elements x and y. By Urysohn’s Lemma, there exists a
function h ∈ C(Y ) such that h(x) = 0 and h(y) = 1. Put f = h ◦ ε and

A = {x ∈ X : f̂(p) − 1/3 ≤ f(x) ≤ f̂(p) + 1/3}. Then A ∈ p by Lemma 5.1, so
x,y ∈ clY (ε(A)), and so h(x),h(y) ∈ clR(f(A)). Therefore, |h(x)−h(y)| ≤ 2/3,
a contradiction.

Since C is a singleton, we obtain a function F : δX → Y . Clearly, F ◦e = ε.
Since e(X) and ε(X) are dense subsets of δX and Y , respectively, it is enough
to show that F is injective and continuous to finish the proof.

To see that F is injective, suppose that p,q ∈ δX satisfy p 6= q. By Urysohn’s
Lemma, there exists a function g ∈ C(δX) such that g(p) = 0 and g(q) = 1.
Put f = g ◦ e. Then f ∈ F by Corollary 5.5. Put A = {x ∈ X : f(x) ≤ 1/3}
and B = {x ∈ X : f(x) ≥ 2/3}. Then A ∈ p and B ∈ q by Lemma 5.1, and
so F(p) ∈ clY (ε(A)) and F(q) ∈ clY (ε(B)). Statement (ii) implies that there
exists a function h ∈ C(Y ) such that f = h◦ ε. Then h(F(p)) ∈ clR(f(A)) and
h(F(q)) ∈ clR(f(B)), and so F(p) 6= F(q), as required.

To show that F is continuous, let p ∈ δX and let U be an open neighborhood
of F(p) in Y with U 6= Y . Again, there exists a continuous function h ∈ C(Y )
such that h(F(p)) = 0 and h(Y \ U) = {1}. Put f = h ◦ ε. The continuity

of h implies that f̂(p) = 0, and so B = {x ∈ X : −1/2 ≤ f(x) ≤ 1/2} ∈ p

by Lemma 5.1. If q ∈ B̂, then h(F(q)) ∈ [−1/2,1/2], and so F(q) ∈ U, thus
finishing the proof. �

6. Some relationships between function lattices

Throughout this section, we assume that F1 and F2 are function lattices on
X contained in ℓ∞(X). We denote by δ1X and δ2X the spaces of F1-ultrafilters
on X and F2-ultrafilters on X, respectively. Also, we denote by e1 and e2 the
canonical mappings from X to δ1X and δ2X, respectively. If A ⊆ X, then the
notation  is ambiguous. However, we hope that it is clear from the notation

used whether we consider  as a subset of δ1X or δ2X. If f ∈ F1 ∩ F2, then
f extends to both δ1X and δ2X. We denote these extension by f

δ1 and fδ2,
respectively.

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T. Alaste

Theorem 6.1. If F1 ⊆ F2, then the following statements are equivalent:

(i) F1 is dense in F2.
(ii) The set {fδ2 : f ∈ F1} is dense in C(δ2X).
(iii) A filter ϕ on X is an F1-filter if and only if ϕ is an F2-filter.
(iv) δ1X = δ2X.

Proof. The equivalence of statements (i) and (ii) and the implication (iv) ⇒ (i)
follow from Theorem 5.4, and the implication (iii) ⇒ (iv) is obvious. To verify
the implication (i) ⇒ (iii), it is enough to show that a non-empty family A of
non-empty subsets of X is an F1-family on X if and only if A is an F2-family
on X. Since F1 ⊆ F2, necessity is obvious, and sufficiency follows from the
density of F1 in F2 and Remark 3.2. �

For the rest of this section, we assume that F1 and F2 are closed subalgebras
of ℓ∞(X) containing the constant functions.

Theorem 6.2. The inclusion F1 ⊆ F2 holds if and only if there exists a
continuous, surjective mapping F : δ2X → δ1X such that e1 = F ◦ e2.

Proof. Suppose first that F1 ⊆ F2. Let p ∈ δ2X and put

C =
⋂

A∈p

clδ1X(e1(A)).

Similar arguments as used in the proof of Theorem 5.7 apply to show that C is
a singleton, and so we obtain a function F : δ2X → δ1X. Clearly, e1 = F ◦ e2.
Also, arguing as in the last part of the proof of Theorem 5.7, we see that F is
continuous. Therefore, we need only to show that F is surjective.

If q ∈ δ1X, then q is an F2-filter on X. Pick any p ∈ δ2X with q ⊆ p
and let A ∈ q. Since δ1X is a regular topological space, there exists a τ(F1)-

open subset B of X such that B ∈ q and clδ1X(B̂) ⊆ Â. Then B ∈ p, so

F(p) ∈ clδ1X(B̂) by Lemma 4.7 (ii), and so A ∈ F(p). Therefore, q ⊆ F(p),
and so q = F(p), as required.

Suppose now that there exists a continuous mapping F : δ2X → δ1X with
e1 = F ◦ e2. Let f ∈ F1. By Theorem 5.2, there exists a function g ∈ C(δ1X)
such that f = g◦e1. Since g◦F ∈ C(δ2X) and f = (g◦F)◦e2, we have f ∈ F2
by Corollary 5.5, thus finishing the proof. �

For the proof of the next theorem, recall the definition of f̂ from the proof
of Theorem 5.2.

Theorem 6.3. Suppose that F1 ⊆ F2 and let F : δ2X → δ1X be as in Theorem
6.2. If p ∈ δ2X and q ∈ δ1X, then the following statements are equivalent:

(i) q ⊆ p.
(ii) F(p) = q.
(iii) fδ2(p) = fδ1(q) for every f ∈ F1.

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Function lattices and compactifications

Proof. (i) ⇒ (ii) This was proved already in the proof of Theorem 6.2.
(ii) ⇒ (iii) Suppose that F(p) = q. Let f ∈ F1. Since e1 = F ◦ e2,

the functions fδ2 and fδ1 ◦ F agree on e2(X), hence, on δ2X. Therefore,
fδ2(p) = fδ1(q).

(iii) ⇒ (i) Suppose that q is not contained in p. Pick some A ∈ q \ p. Pick
B ∈ q and a positive function f ∈ F1 with f(B) = {0} and f(X \ A) = {1}.
Then fδ1(q) = 0. Since X(f,1/2) /∈ p, there exists some C ∈ p such that
X(f,1/3) ∩ C = ∅ by Theorem 3.9 (ii). Then fδ2(p) ≥ 1/3, thus finishing the
proof. �

Define two closed equivalence relations ∼ and ≈ on δ2X as follows: p ∼ q
if and only if F(p) = F(q), and p ≈ q if and only if fδ2(p) = fδ2(q) for every
f ∈ F1. Theorem 6.3 shows that these relations are identical. Since F is a
quotient mapping (see [16, pp. 60-61]), we obtain the following statement.

Corollary 6.4. If F1 ⊆ F2, then the quotient space δ2X/ ≈ is homeomorphic
with δ1X.

7. F-filters and ideals of F

Throughout this section, we assume that F is a closed subalgebra of ℓ∞(X)
containing the constant functions. We establish a correspondence between F-
filters on X and closed, proper ideals of F. Roughly speaking, we show how
the ideals of F can be used to generate F-filters on X. We apply the following
convention for the rest of this paper: By an ideal of F, we always mean a
closed, proper ideal of F.

The next lemma follows from [7, (1.23) Proposition]. Since the proof of the
cited proposition relies on the spectrums of single elements of C∗-algebras, we
present the following short proof using only basic properties of Banach algebras.

Lemma 7.1. If f ∈ F \ F0, then 1/f ∈ F.

Proof. Suppose first that f ∈ F \ F0 is positive. Pick r > 0 such that r ≤ f(x)
for every x ∈ X. Then

0 <
r

‖f‖
≤
f(x)

‖f‖
≤ 1

for every x ∈ X. Put g = f/‖f‖. Then ‖1 − g‖ < 1 by the inequalities above,
and so g is invertible in F (see [7, (1.3) Lemma]). Therefore, 1/f ∈ F.

If f ∈ F \ F0 is any function, then f
2 ∈ F \ F0 is positive. The equality

1/f = f/f2 and the first part of the proof imply that 1/f ∈ F. �

Corollary 7.2. If I is an ideal of F, then I ⊆ F0.

Definition 7.3. For every ideal I of F, define

(7.1) B(I) = {X(f,r) : f ∈ I, r > 0}.

Theorem 7.4. If I is an ideal of F, then B(I) is a filter base on X and the
filter ϕ on X generated by B(I) is an F-filter. Conversely, if ϕ is an F-filter
on X, then there exists an ideal I of F such that ϕ is generated by B(I).

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 197



T. Alaste

Proof. Suppose first that I is an ideal of F. First, X(f,r) 6= ∅ for every f ∈ I
and for every r > 0 by Corollary 7.2, and so ∅ /∈ B(I). Next, let f,g ∈ I and
let r > 0. Since f2 +g2 ∈ I and X(f2 +g2,r) ⊆ X(f,r) ∩X(g,r), the set B(I)
is a filter base on X. Since B(I) is an F-family on X by Lemma 3.5, the filter
ϕ on X generated by B(I) is an F-filter.

Suppose now that ϕ is an F-filter on X. Put

I = {f ∈ F : X(f,r) ∈ ϕ for every r > 0}.

Clearly, 0 ∈ I. Let f1,f2 ∈ I, let h ∈ F with h 6= 0, let α ∈ R with α 6= 0,
let (gn) be a sequence in I converging to some g ∈ F, and let r > 0. The
inclusions

X(f1,r/2) ∩ X(f2,r/2) ⊆ X(f1 − f2,r),

X(f1,r/‖h‖) ⊆ X(f1h,r),

X(f1,r/|α|) ⊆ X(αf1,r),

X(gn,r/2) ⊆ X(g,r),

where the last one holds if ‖gn − g‖ ≤ r/2, imply that I is an ideal of F.
We claim that B(I) is a filter base for ϕ. Clearly, B(I) ⊆ ϕ, so suppose

that A ∈ ϕ satisfies A 6= X. Pick some B ∈ ϕ and a function f ∈ F with
f(B) = {0} and f(X \A) = {1}. Since B ⊆ X(f,r) for every r > 0 and B ∈ ϕ,
we have f ∈ I. Since X(f,1/2) ⊆ A, the claim follows. �

Let ϕ be an F-filter on X. The previous theorem guarantees the existence
of an ideal I of F such that ϕ is generated by B(I). The next theorem shows
that this ideal I is unique.

Theorem 7.5. Let I be an ideal of F, let ϕ be the F-filter on X generated by
B(I), and let f ∈ F. The following statements are equivalent:

(i) f ∈ I.

(ii) f̂(p) = 0 for every p ∈ ϕ.
(iii) X(f,r) ∈ ϕ for every r > 0.

Proof. (i) ⇒ (ii) Suppose that f ∈ I. Let p ∈ ϕ and let r > 0. Since ϕ is

generated by B(I), we have X(f,r) ∈ p by Theorem 4.11, and so |f̂(p)| ≤ r by

Lemma 4.6. Therefore, f̂(p) = 0.
(ii) ⇒ (iii) This follows from Lemma 5.1 and Theorem 4.2 (ii).
(iii) ⇒ (i) Suppose that (iii) holds. It is enough to show that f ∈ clF(I),

and so we may assume that f 6= 0. Let 0 < r < ‖f‖. Then X(f,r) 6= X.
Since B(I) is a filter base for ϕ, there exist functions h ∈ I and g ∈ F such
that g(X(h,1)) = {0} and g(X \ X(f,r)) = {1}. Now, 1/(|h| ∨ 1)2 ∈ F
by Lemma 7.1, so k := h2/(|h| ∨ 1)2 ∈ I, and so fk ∈ I. The inclusion
X(h,1) ⊆ X(f,r) implies that f and fk agree on X \ X(f,r). Therefore,
‖f − fk‖ = supx∈X(f,r)|f(x)(1 − k(x))| ≤ r, and so f ∈ clF(I), thus finishing
the proof. �

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Function lattices and compactifications

Let ϕ be an F-filter on X. We say that a function f ∈ F tends to zero in
the direction of ϕ if and only if, for every r > 0, there exists some A ∈ ϕ with
|f(x)| ≤ r for every x ∈ A. The previous theorem, then, says that a subset I
of F is an ideal of F if and only if there exists an F-filter ϕ on X such that I
consists of those members of F which tend to zero in the direction of ϕ.

Let I be an ideal of F. The equality X(f,r) = X(|f|,r) for every f ∈ F
and for every r > 0 implies that |f| ∈ I for every f ∈ I.

For every ideal I of F, we denote by ϕ(I) the F-filter on X generated by
B(I). If I and J are ideals of F, then the previous theorem implies that I ⊆ J
if and only if ϕ(I) ⊆ ϕ(J). From this we conclude the following: An ideal I of
F is a maximal ideal of F if and only if ϕ(I) is an F-ultrafilter on X.

We denote by M(F) the set of all maximal ideals of F. If I is an ideal of F,
then the hull of I is the set h(I) = {J ∈ M(F) : I ⊆ J}. The kernel k(J ) of
a non-empty subset J of M(F) is the set k(J ) =

⋂
J∈J

J. Note that k(J ) is
an ideal of F. The hull-kernel topology on M(F) is defined by declaring a non-
empty subset J of M(F) to be closed if and only if J = h(k(J )). In terms of
F-filters, this reads as follows: A non-empty subset J of M(F) is closed if and
only if there exists an F-filter ϕ on X such that J = {J ∈ M(F) : ϕ ⊆ ϕ(J)}.
Therefore, the mapping J 7→ ϕ(J) from M(F) to δX is a homeomorphism.

The following well-known property of F follows from Theorem 4.2 (ii).

Corollary 7.6. If I is an ideal of F, then k(h(I)) = I.

8. F-filters on topological spaces

In the previous sections, we made no assumption about algebraic or topo-
logical structure on the set X. In this section, we assume that (X,τ) is a
Hausdorff topological space and that F is a function lattice on X such that
F ⊆ C(X).

Recall that A◦ denotes the τ(F)-interior of a subset A of X. If A ⊆ X,

then e−1(Â) = A◦. Since F ⊆ C(X), the set A◦ is τ-open in X, and so the
canonical mapping e : X → δX is continuous. For every element x ∈ X, we
denote by N(x) the neighborhood filter of x in (X,τ). Since F ⊆ C(X), we
have NF(x) ⊆ N(x) for every x ∈ X.

The canonical mapping e : X → δX is an embedding if and only if the
equality N(x) = NF(x) holds for every x ∈ X. By Remark 3.5, this equality
for every x ∈ X is equivalent to statement (ii) below.

Lemma 8.1. The following statements are equivalent:

(i) The canonical mapping e : X → δX is an embedding.
(ii) For every x ∈ X and for every neighborhood U ∈ N(x) with U 6= X,

there exists a function f ∈ F with f(x) = 1 and f(X \ U) = {0}.

The next theorem follows from Theorem 5.7.

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T. Alaste

Theorem 8.2. If Y is a compact Hausdorff space and ε : X → Y is a con-
tinuous mapping such that ε(X) is dense in Y , then the following statements
hold:

(i) The set F = {h◦ε : h ∈ C(Y )} is a closed subalgebra of C(X) containing
the constant functions.

(ii) F is isometrically isomorphic with C(Y ).
(iii) There exists a homeomorphism F : δX → Y such that F ◦ e = ε.

Statements (ii) and (iii) of the next corollary constitute Stone-Weierstrass
Theorem.

Corollary 8.3. If X is compact, then the following statements are equivalent:

(i) The canonical mapping e : X → δX is a homeomorphism.
(ii) F separates the points of X.
(iii) F is dense in C(X).

Proof. Since X is compact, the canonical mapping e : X → δX is a continuous
surjection. Therefore, e is a homeomorphism if and only if e is injective, and
so (i) and (ii) are equivalent.

(i) ⇒ (iii) Suppose that e : X → δX is a homeomorphism. Then it is easy
to verify that the mapping g 7→ g ◦ e from C(δX) to C(X) is an isometric

isomorphism. Since {f̂ : f ∈ F} is dense in C(δX) by Theorem 5.4 and

f̂ ◦ e = f for every f ∈ F, the statement follows.
(iii) ⇒ (ii) This follows from Urysohn’s Lemma. �

Next statement is a consequence of the Gelfand-Naimark Theorem. Here,
it follows from Corollary 8.3. An isometric isomorphism T : C(X) → C(Y )
induces a bijection between ideals of C(X) and C(Y ). Then the maximal ideal
spaces M(C(X)) and M(C(Y )) are homeomorphic (under their hull-kernel
topologies).

Corollary 8.4. If X and Y are compact Hausdorff spaces, then X and Y are
homeomorphic if and only if C(X) and C(Y ) are isometrically isomorphic.

We finish this section with the following statement concerning locally com-
pact topological spaces. If X is locally compact, then we denote by X∞ the
one-point compactification of X. Let e1 : X → X∞ denote the natural embed-
ding. Then {h ◦ e1 : h ∈ C(X∞)} = C0(X) ⊕ R, where R denotes the constant
functions on X. Necessity of the following statement follows from Corollary
5.5 using the zero extension. Sufficiency follows from Theorem 6.2 and from
the fact that X is embedded and open in X∞.

Theorem 8.5. Suppose that X is non-compact and locally compact and that F
is a closed subalgebra of C(X) containing the constant functions. The canonical
mapping e : X → δX is an embedding and e(X) is open in δX if and only if
C0(X) ⊆ F.

Remark 8.6. Let X and F be as above and suppose that X is embedded in
δX. Then the family ϕK = {X \ K : K ⊆ X and clX(K) is compact} is an

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Function lattices and compactifications

F-filter on X and δX \ e(X) = ϕ̂K. In particular, if δX = X∞, then ∞ = ϕK
as an F-filter.

9. Spectrums of unital C∗-subalgebras of ℓ∞(X)

In this last section, we change our notation from spaces of real-valued func-
tions to spaces of complex-valued functions. We denote by ℓ∞(X) the C∗-
algebra of all bounded, complex-valued functions on X. If X is a topological
space, then we denote by C(X) the C∗-subalgebra of ℓ∞(X) consisting of con-
tinuous members of ℓ∞(X). We explain briefly how the introduced filters can
be used to represent the spectrum of any C∗-subalgebra of ℓ∞(X) as a space
of filters on X.

Throughout this section, let F be a C∗-subalgebra of ℓ∞(X) such that F
contains the constant functions. We consider the spectrum ∆ of F as the space
of all non-zero, multiplicative linear functionals on F, that is,

∆ = {µ ∈ F∗ : µ 6= 0 and µ(fg) = µ(f)µ(g) for all f,g ∈ F},

where F∗ denotes the Banach dual of F. The evaluation mapping ε : X → ∆
is defined by [ε(x)](f) = f(x) for every x ∈ X and for every f ∈ F.

Under the relative weak* topology of F∗, the space ∆ is a compact Hausdorff
space and ε(X) is a dense subset of ∆. The characteristic property of the space
∆ is the fact that F and C(∆) are isometrically ∗-isomorphic. If µ ∈ ∆, then
kerµ = {f ∈ F : µ(f) = 0} is a maximal ideal of F. Conversely, if I is
a maximal ideal of F, then there exists a unique element µ ∈ ∆ such that
I = kerµ (see [7]).

The space Fr of all real-valued members of F is a closed subalgebra of the
space of all bounded, real-valued functions on X, and so Fr is a function lattice
on X. We define δX to be the space of all Fr-ultrafilters on X. If f ∈ F, then
Theorem 5.2 implies that the real and imaginary parts of f extend to δX, and

so there exists a unique function f̂ ∈ C(δX) satisfying f = f̂ ◦ e. Then the

mapping f 7→ f̂ from F to C(δX) is an isometric ∗-isomorphism by Theorem
5.4.

Small adjustments in Section 7 apply to show that, for every F-filter ϕ on
X, there exists a unique ideal I of F such that ϕ is generated by B(I). (Here,
B(I) is defined as in (7.1).) In the second part of the proof of Lemma 7.1,

we apply the equality 1/f = f/|f|2. Here, f(x) = f(x) for every x ∈ X and

f(x) denotes the complex-conjugate of f(x). In the last part of the proof of
Theorem 7.4, we apply the fact that |f|2 + |g|2 ∈ I. In the proof of implication
(iii) ⇒ (i) of Theorem 7.5, we define k = |h|2/(|h| ∨ 1)2. Here, we apply the
fact that |f| ∈ F for every f ∈ F (see [11, p. 265]). If I is an ideal of F, then
the equalities X(f,r) = X(|f|,r) = X(f,r), which hold for every f ∈ F and
for every r > 0, and Theorem 7.5 imply that |f| ∈ I and f ∈ I for every f ∈ I.

Finally, the mapping µ 7→ kerµ from ∆ to the maximal ideal space M(F)
of F is a bijection. Once M(F) is equipped with the hull-kernel topology, this
mapping is a homeomorphism. Since the mapping J 7→ ϕ(J) from M(F) to

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T. Alaste

δX, where ϕ(J) is the F-ultrafilter on X generated by B(J), is a homeomor-
phism, we conclude that the mapping µ 7→ p(µ) from ∆ to δX, where p(µ) is
the F-ultrafilter on X generated by B(kerµ), is a homeomorphism.

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